To inscribe a square in a given circle.

Let *ABCD* be the given circle.

It is required to inscribe a square in the circle *ABCD.*

Draw two diameters *AC* and *BD* of the circle *ABCD* at right angles to one another, and join *AB, BC, CD,* and *DA.*

Then, since *BE* equals *ED,* for *E* is the center, and *EA* is common and at right angles, therefore the base *AB* equals the base *AD.*

For the same reason each of the straight lines *BC* and *CD* also equals each of the straight lines *AB* and *AD.* Therefore the quadrilateral *ABCD* is equilateral.

I say next that it is also right-angled.

For, since the straight line *BD* is a diameter of the circle *ABCD,* therefore *BAD* is a semicircle, therefore the angle *BAD* is right.

For the same reason each of the angles *ABC, BCD,* and *CDA* is also right. Therefore the quadrilateral *ABCD* is right-angled.

But it was also proved equilateral, therefore it is a square, and it has been inscribed in the circle *ABCD.*

Therefore the square *ABCD* has been inscribed in the given circle.

Q.E.F.